Clearly, the spectrum \(\sigma (A^\varepsilon )\)  belongs to the closed
real positive semi-axis \([0,+\infty )=\overline{{\mathbb {R}}_+}\subset {\mathbb {C}}\) . Moreover, according to the
Floquet–Bloch–Gelfand theory, see for instance
[1]}, [2]}, [3]}, [4]}, [5]}, the spectrum gets
the band-gap structure
\(\begin{array}{c}\sigma (A^\varepsilon )=\bigcup \limits _{p\in {\mathbb {N}}}\beta _p^\varepsilon \, ,\end{array}\) 
